The description of symmetries via defects is particularly convenient in the context
of orbifold theories.Orbifolds of two-dimensional conformal field theories
with respect to an outer automorphism group symmetry were studied
long ago, see for example [1]}, [2]}.
The partition function of the orbifold theory
can be computed by evaluating the original theory on a sufficiently fine defect network
labeled with symmetry defects.By `sufficiently fine' we mean
a defect network that can be obtained as the dual graph of a
triangulation of \(\Sigma \) . This requires the introduction of point defects
— called `junction fields' —
on which three defect lines can meet.
We use the defect approach to orbifolds [3]}, [4]}, [5]}, [6]}
in Section REF 
to show that the orbifold theory is the Yang-Mills theory on \(\Sigma \)  based on the gauge group
\(G\rtimes \mathsf {Out}(G)\) . An advantage of the defect approach to orbifolds is that it
also works for suitable non-invertible defects. We use this generalised orbifold
construction to show in Section REF  that the orbifold of the Yang-Mills theory with gauge
group \(G\rtimes \mathsf {Out}(G)\) , with respect to the Wilson line defects induced from
the regular
representation \(L^2(\mathsf {Out}(G))\)  of \(\mathsf {Out}(G)\) ,
is the Yang-Mills theory on \(\Sigma \)  with gauge group
\(G\) ; these defects are invertible, and hence correspond to symmetry defects, only when
\(\mathsf {Out}(G)\)  is an abelian group. This may be thought of as a duality between these two quantum Yang-Mills theories with defects.
